Orders of Vanishing and U(1) Charges in F-theory
arXiv:2110.10159 · doi:10.1007/JHEP03(2022)051
Abstract
Many interesting questions about F-theory models, including several concerning the F-theory swampland, involve massless matter charged under U(1) gauge symmetries. It is therefore important to better understand the geometric properties of F-theory models realizing various U(1) charges. We propose that, for F-theory models described by elliptic fibrations in Weierstrass form, the U(1) charge of light matter is encoded in the orders of vanishing of the section components corresponding to the U(1) gauge symmetry. We give specific equations relating the U(1) charges to the orders of vanishing that seem to hold for both U(1)-charged singlets and for matter additionally charged under a simply-laced nonabelian gauge algebra. Our formulas correctly describe properties of F-theory models in the prior literature, and we give an argument that they should describe the orders of vanishing for arbitrarily high U(1) charges. They also resemble formulas for the -adic valuations of elliptic divisibility sequences developed by Stange [arXiv:1108.3051]. These proposals could serve as a U(1) analogue of the Katz-Vafa method, allowing one to determine U(1) charges without resolution. Additionally, they predict geometric information about F-theory models with general U(1) charges, which may be useful for exploring the F-theory landscape and swampland.
112 pages with appendices; v2: references added
References in corpus (20)
- Les Houches Lectures on Constructing String Vacua
- T-Branes and Monodromy
- F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds
- F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
- Global aspects of the space of 6D N = 1 supergravities
- TASI Lectures on Supergravity and String Vacua in Various Dimensions
- A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
- Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group
- General Prescription for Global U(1)'s in 6D SCFTs
- F-GUTs with Mordell-Weil U(1)'s
- Tall sections from non-minimal transformations
- Generic matter representations in 6D supergravity theories
- Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces
- Exotic matter on singular divisors in F-theory
- Tate's Algorithm for F-theory GUTs with two U(1)s
- Automatic Enhancement in 6D Supergravity and F-theory Models
- On genus one fibered Calabi-Yau threefolds with 5-sections
- Half-hypermultiplets and incomplete/complete resolutions in F-theory
- Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s
- Matter representations from geometry: under the spell of Dynkin